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A361216 Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X k rectangle. 7
1, 1, 4, 2, 11, 56, 3, 29, 370, 5752, 4, 94, 2666, 82310, 2519124, 6, 263, 19126, 1232770, 88117873, 6126859968, 12, 968, 134902, 19119198, 2835424200 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Tilings that are rotations or reflections of each other are considered distinct.
Pieces can have any combination of integer side lengths, but for the optimal sets computed so far (up to (n,k) = (7,5)), all pieces have one side of length 1.
LINKS
FORMULA
T(n,1) = A102462(n).
EXAMPLE
Triangle begins:
n\k| 1 2 3 4 5 6 7 8
---+--------------------------------------------------------
1 | 1
2 | 1 4
3 | 2 11 56
4 | 3 29 370 5752
5 | 4 94 2666 82310 2519124
6 | 6 263 19126 1232770 88117873 6126859968
7 | 12 968 134902 19119198 2835424200 ? ?
8 | 20 3416 1026667 307914196 109979838540 ? ? ?
A 3 X 3 square can be tiled by three 1 X 2 pieces and three 1 X 1 pieces in the following ways:
+---+---+---+ +---+---+---+ +---+---+---+
| | | | | | | | | | | |
+---+---+---+ + +---+---+ +---+ +---+
| | | | | | | | | | |
+---+---+ + +---+---+ + +---+---+ +
| | | | | | | | |
+---+---+---+ +---+---+---+ +---+---+---+
.
+---+---+---+ +---+---+---+ +---+---+---+
| | | | | | | | | |
+---+---+---+ +---+---+ + +---+---+---+
| | | | | | | | | |
+---+---+ + +---+---+---+ +---+---+---+
| | | | | | | | |
+---+---+---+ +---+---+---+ +---+---+---+
.
+---+---+---+ +---+---+---+
| | | | | |
+---+---+---+ +---+---+---+
| | | | | |
+---+---+---+ +---+---+---+
| | | | | |
+---+---+---+ +---+---+---+
The first six of these have no symmetries, so they account for 8 tilings each. The last two has a mirror symmetry, so they account for 4 tilings each. In total there are 6*8+2*4 = 56 tilings. This is the maximum for a 3 X 3 square, so T(3,3) = 56.
The following table shows the sets of pieces that give the maximum number of tilings up to (n,k) = (7,5). The solutions are unique except for (n,k) = (2,1) and (n,k) = (6,1).
\ Number of pieces of size
(n,k)\ 1 X 1 | 1 X 2 | 1 X 3 | 1 X 4
------+-------+-------+-------+------
(1,1) | 1 | 0 | 0 | 0
(2,1) | 2 | 0 | 0 | 0
(2,1) | 0 | 1 | 0 | 0
(2,2) | 2 | 1 | 0 | 0
(3,1) | 1 | 1 | 0 | 0
(3,2) | 2 | 2 | 0 | 0
(3,3) | 3 | 3 | 0 | 0
(4,1) | 2 | 1 | 0 | 0
(4,2) | 4 | 2 | 0 | 0
(4,3) | 3 | 3 | 1 | 0
(4,4) | 5 | 4 | 1 | 0
(5,1) | 3 | 1 | 0 | 0
(5,2) | 4 | 3 | 0 | 0
(5,3) | 4 | 4 | 1 | 0
(5,4) | 7 | 5 | 1 | 0
(5,5) | 7 | 6 | 2 | 0
(6,1) | 2 | 2 | 0 | 0
(6,1) | 1 | 1 | 1 | 0
(6,2) | 4 | 4 | 0 | 0
(6,3) | 7 | 4 | 1 | 0
(6,4) | 8 | 5 | 2 | 0
(6,5) | 10 | 7 | 2 | 0
(6,6) | 11 | 8 | 3 | 0
(7,1) | 2 | 1 | 1 | 0
(7,2) | 5 | 3 | 1 | 0
(7,3) | 8 | 5 | 1 | 0
(7,4) | 10 | 6 | 2 | 0
(7,5) | 11 | 7 | 2 | 1
CROSSREFS
Main diagonal: A361217.
Columns: A102462 (k = 1), A361218 (k = 2), A361219 (k = 3), A361220 (k = 4).
Sequence in context: A182870 A094406 A142706 * A092952 A286145 A010318
KEYWORD
nonn,tabl,more
AUTHOR
STATUS
approved

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Last modified April 19 08:28 EDT 2024. Contains 371782 sequences. (Running on oeis4.)